The Fourier transform on an LCA group

Let $G$ be a locally compact Hausdorff abelian group, written additively, let $m_G$ be a fixed left Haar measure on $G$ (Left Haar integral and left Haar measure), and let $\widehat{G}$ be the Pontryagin dual with the compact-open topology (The Pontryagin dual with the compact-open topology).

For $f\in L^1(G,m_G)$ (The space $L^p(\mu )$ as the quotient by null functions, Integrable real and complex functions, and their integrals) the Fourier transform of $f$ is the function $\widehat{f}:\widehat{G}\to \mathbb{C}$ defined at $\gamma \in \widehat{G}$ by $\widehat{f}(\gamma ):={\int }_{G}f(x)\overline{\gamma (x)}d{m}_G(x)\mathrm{.}$

Well-definedness. The evaluation pairing $(\gamma ,x)\mapsto \gamma (x)$ is jointly continuous (Evaluation of characters is jointly continuous) and every character takes values in the unit circle $\mathbb{T}=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$ (The multiplicative unit circle is a compact metrizable topological abelian group), so for fixed $\gamma$ the function $x\mapsto \overline{\gamma (x)}$ is Borel measurable with modulus $1$ (Composition with a Borel measurable outer map preserves measurability, Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive). Hence the integrand $x\mapsto f(x)\overline{\gamma (x)}$ is Borel measurable and dominated by $\mathrm{\mid }f\mathrm{\mid }$, so the integral converges absolutely and $\mathrm{\mid }\widehat{f}(\gamma )\mathrm{\mid }\le {\int }_G\mathrm{\mid }f\mathrm{\mid }d{m}_G=\mathrm{\parallel }f{\mathrm{\parallel }}_1\text{for every }\gamma \mathrm{.}$ Replacing $f$ by an $m_G$-a.e. equal representative changes the integrand only on an $m_G$-null set, so no value $\widehat{f}(\gamma )$ changes: the transform is well defined on the quotient $L^1(G,m_G)$ and not merely on representatives, and $f\mapsto \widehat{f}$ is a linear map $L^1(G,m_G)\to {\mathrm{\ell }}^{\mathrm{\infty }}(\widehat{G})$ with $\mathrm{\parallel }\widehat{f}{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }f{\mathrm{\parallel }}_1$.

Convention. This is the conjugate-phase convention. On $G={\mathbb{R}}^n$ with Lebesgue measure, where the characters are ${\gamma }_{\xi }(x)=e^{2\pi ix\cdot \xi }$, it reads $\widehat{f}(\xi )={\int }_{{\mathbb{R}}^n}f(x)e^{-2\pi ix\cdot \xi }dx$. No dual Haar measure is used in the definition: the compatible scale on $\widehat{G}$ is fixed only by the compatible dual Haar normalisation proved on this page.